## Introduction to Knot Theory

December 7th, 2015

This site contains information on the lecture “Knots” given in autumn 2015 at Cardiff University, School of Mathematics.

## News and Announcements

*— none currently —*

## Organisational matters

## Resources

Many resources on knot theory exist, from classical textbooks over lecture notes to picture galleries, videos and software. Here I list a few of them.

**Books**

- Colin Adams:
*The Knot Book*(Freeman and Company, 2001)

An easy to read elementary introduction to knot theory. Gives good explanations of many of the relevant concepts, but does not cover all the material of this course. Still highly recommended as a first reading. - Charles Livingston:
*Knot Theory*(Math. Ass. of America, 1993)

Another very good introduction to knot theory, more mathematical than the one by Adams. - Peter Cromwell (Cambridge University Press, 2004) Another good introduction to knot theory, more mathematical than the one by Livingston – often more technical than the style of this lecture will be.
- Louis Kauffman: Knots and Physics (World Scientific, 1991) A large but very readable book on theory. Most of the topics covered in this course can be found in Part I. We will not touch the material in Part II.

**Lecture Notes**

- A knot theory course at Warwick (2007), by Brian Sanderson from the University of Warwick.
- Knot Knotes (2015), by Justin Roberts from the University of San Diego.

**Websites**

- The Knot Atlas

A user-editable atlas of knots with many tables of knots, such as this one. - A Knot Zoo

Another table of knots and links. - Knots on the Web

An extensive list of web resources on many topics related to knots.

**Videos**

- Unraveling of a complicated looking knot into the unknot.
- Animation of a Brunnian link.
- A youtube channel with some introductory videos on knot theory, by Carlo H. Séquin from UC Berkeley.

**Software**

- KnotPlot A versatile programme for visualising knots and manipulating them, including some functions to compute mathematical invariants and several educational demonstrations. Has a free trial version. The website also hosts many pictures of knots in space — all pictures of knots displayed on this site are made with knotplot.
- A collection of knot theory demonstrations, interactive with the Wolfram CDF Player.
- The Mathematica Knot Theory Package. See also here.

## Exercises

Here you can download the exercise sheets of this course.

- Exercise Sheet 1 (to be discussed in class on 9 October)
- Exercise Sheet 2 (to be discussed in class on 23 October)
- Exercise Sheet 3 (to be discussed in class on 6 November)
- Exercise Sheet 4 (to be discussed in class on 4 December)
- Exercise Sheet 5 (to be discussed in class on 20 November)

In addition to the exercises, I also offer some *quizzes* consisting of quick questions for you to test your understanding of the material presented in the lecture:

## Lectures

Here you find a list of all the topics covered in the lecture. This list just serves as a reminder of what has been discussed, it is not a replacement for a book/lecture notes.

**Lecture 1: Introduction, geometrical knots****Lecture 2: Topological knots/links and link invariants****Lecture 3: Link diagrams****Lecture 4: Reidemeister moves****Lecture 5: Reidemeister’s Theorem. 3-Colourability****Lecture 6: 3-Colourability and p-Colourability****Lecture 7: The determinant of a link****Lecture 8: The Alexander Polynomial****Lecture 9: The Colouring Group****Lecture 10: The Bracket Polynomial****Lecture 11: The Kauffman Bracket****Lecture 12: The normalized bracket polynomial****Lecture 13: The Jones polynomial****Lecture 14: Composite and prime knots****Lecture 15: The HOMFLY and Conway polynomials****Lecture 16: Graphical calculus for matrices****Lecture 17: Matrices and link diagrams****Lecture 18: The partition function****Lecture 19: A matrix model for the Kauffman bracket****Lecture 20: The braid group****Lecture 21: Alexander’s and Markov’s Theorems****Lecture 22: The Jones polynomial and the braid group**